[bigpooch]

Technically, if n is odd, you won't have "n/2 pairs".

A very simple solution, one that most children understand,
is based on the commutativity of addition and is really the
same idea:

1+2+3+...+n = S
n+(n-1)+...+1 = S

Adding the two equations above,

(n+1)+...+(n+1) = 2S

or n(n+1) = 2S
or S = n(n+1)/2.


IDEAS
=====

An idea can be very simple (such as above) or complex (such
as Kummer's idea of numbers of the form a+bi in attempt to
solve FLT). Even when an idea fails to solve a problem, it
can create a body of theory. What differentiates great
problem solvers from ordinary ones is not only perseverance,
but finding the "key ideas". Often, solving problems
require many ideas; that is why mathematicians often need
to have exposure to analysis and abstract algebra as well as
their area(s) of research.

Also, some ideas for proofs are very beautiful, but often
require some background. For example, I am still enamored
with Furstenberg's proof of the infinitude of primes (based
on point-set topology), but obviously it is not as easily
accessible as the common proof attributed to Euclid.